Chandrasekhar Hall

Local Cohomology and Completion

Rishi Vyas

Ben Gurion University, Israel

Fix an ideal in a commutative noetherian ring. There is an interestingrelationship between the Local Cohomology and Completion functors at thisideal: their (right and left, respectively) derived functors are adjoint toeach other, and this adjunction induces an interesting equivalence ofcategories. This relationship has been independently studied by a number ofmathematicians over the last few decades: protagonists for this storyinclude Grothendieck, Matlis, Greenlees-May, Alonso-Jeremias-Lipman,Schenzel, Porta-Shaul-Yekutieli, and Positselski.

In this expository talk, we will try and describe this relationship byfocusing on the simplest example: an ideal generated by a single element.The only prerequisites required are a basic understanding of commutativealgebra, and some, again basic, familiarity with chain complexes of modulesover a commutative ring. In particular, most of this talk will not assumeany familiarity with the derived category of a ring.